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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1008.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1008.d1 | 1008h4 | \([0, 0, 0, -2691, 53730]\) | \(1443468546/7\) | \(10450944\) | \([2]\) | \(512\) | \(0.54727\) | |
| 1008.d2 | 1008h3 | \([0, 0, 0, -531, -3726]\) | \(11090466/2401\) | \(3584673792\) | \([2]\) | \(512\) | \(0.54727\) | |
| 1008.d3 | 1008h2 | \([0, 0, 0, -171, 810]\) | \(740772/49\) | \(36578304\) | \([2, 2]\) | \(256\) | \(0.20070\) | |
| 1008.d4 | 1008h1 | \([0, 0, 0, 9, 54]\) | \(432/7\) | \(-1306368\) | \([2]\) | \(128\) | \(-0.14587\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1008.d have rank \(1\).
Complex multiplication
The elliptic curves in class 1008.d do not have complex multiplication.Modular form 1008.2.a.d
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
